Ingeniería Biomédica
2026-01-29
A signal is a function that conveys information by mapping an independent variable (or variables) to measurable quantities.
Formally, a signal is a mapping \(x:\ \mathcal{T}\rightarrow\mathcal{A}\), where \(\mathcal{T}\) is the domain (e.g., time \(t\in\mathbb{R}\) or sample index \(n\in\mathbb{Z}\)) and \(\mathcal{A}\) is the codomain (e.g., amplitudes in \(\mathbb{R}\) or \(\mathbb{C}\), vectors, or matrices).
A deterministic signal \(x_d(t)\) is completely specified by an explicit rule; it produces the same waveform on every observation (e.g., \(x_d(t)=A\cos(2\pi f_0 t+\phi)\)).
A random signal (stochastic process) \(X(t)\) is a family of random variables indexed by \(t\); each observation yields a different realization. It is characterized statistically by its mean \(\mu_X(t)\) and autocorrelation \(R_X(\tau)\).
Even
\[f\left(t\right) = f\left(-t\right)\] \[f\left[t\right] = f\left[-t\right]\]
Odd
\[f\left(t\right) = -f\left(-t\right)\] \[f\left[t\right] = -f\left[-t\right]\]
Decomposition
All signal can be decomposed in two signals: one even, one odd.
\[x(t) = x_{even}(t) + x_{odd}(t)\]
Where:
\[x_{even}(t) = \frac{x(t)+x(-t)}{2} \] \[x_{odd}(t) = \frac{x(t)-x(-t)}{2} \]
Example
Decompose the signal \(x(t)=e^{t}\) into its even and odd parts
\[x_{\text{even}}(t) = \frac{x(t) + x(-t)}{2}\]
\[x_{\text{odd}}(t) = \frac{x(t) - x(-t)}{2}\]
\[x(-t) = e^{-t}\]
\[x_{\text{even}}(t) = \frac{e^t + e^{-t}}{2} = \cosh(t)\]
\[x_{\text{odd}}(t) = \frac{e^t - e^{-t}}{2} = \sinh(t)\]
\[x(t) = x_{\text{even}}(t) + x_{\text{odd}}(t)\]
\[e^t = \cosh(t) + \sinh(t)\]
Definitions. For a signal \(x(t)\) (continuous-time, CT) or \(x[n]\) (discrete-time, DT):
Energy: \(E=\int_{-\infty}^{\infty}\lvert x(t)\rvert^2,dt\) (CT), \(\quad E=\sum_{n=-\infty}^{\infty}\lvert x[n]\rvert^2\) (DT).
Average power: \(P=\lim_{T\to\infty}\dfrac{1}{2T}\int_{-T}^{T}\lvert x(t)\rvert^2,dt\) (CT), \(\quad P=\lim_{N\to\infty}\dfrac{1}{2N+1}\sum_{n=-N}^{N}\lvert x[n]\rvert^2\) (DT).
Energy signal: \(0<E<\infty\) and \(P=0\) (e.g., \(x_E(t)=e^{-a t}u(t)\), \(a>0\), with \(E=\tfrac{1}{2a}\)).
Power signal: \(0<P<\infty\) and \(E=\infty\) (e.g., \(x_P(t)=\cos(2\pi f_0 t)\), with \(P=\tfrac{1}{2}\)).
In signal processing, the classification of a signal \(x(t)\) depends on its behavior over an infinite time interval.
Total Energy (\(E\)) \[E = \int_{-\infty}^{\infty} |x(t)|^2 dt\]
Average Power (\(P\)) \[P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt\]
A signal is classified as an energy signal if and only if: \[0 < E < \infty\] This condition implies that the average power \(P = 0\).
Consider the signal \(x(t) = e^{-at} u(t)\) where \(a > 0\).
Proof of Energy: \[E = \int_{0}^{\infty} (e^{-at})^2 dt = \int_{0}^{\infty} e^{-2at} dt\]
Evaluating the definite integral: \[E = \left[ \frac{e^{-2at}}{-2a} \right]_{0}^{\infty} = 0 - \left( \frac{1}{-2a} \right) = \frac{1}{2a}\]
Conclusion: Since \(E\) is a finite value, it is an energy signal.
A signal is classified as a power signal if and only if: \[0 < P < \infty\] This condition implies that the total energy \(E = \infty\).
Consider \(x(t) = A \cos(\omega_0 t)\).
Proof of Power: Using the period \(T = \frac{2\pi}{\omega_0}\): \[P = \frac{1}{T} \int_{0}^{T} |A \cos(\omega_0 t)|^2 dt = \frac{A^2}{T} \int_{0}^{T} \frac{1 + \cos(2\omega_0 t)}{2} dt\]
The integral of the oscillating term over a full period is zero: \[P = \frac{A^2}{2T} [t]_0^T = \frac{A^2}{2}\]
Conclusion: Power is finite (\(P = A^2/2\)), thus it is a power signal.
| Signal Type | Energy (\(E\)) | Power (\(P\)) | Characteristics |
|---|---|---|---|
| Energy | \(0 < E < \infty\) | \(P = 0\) | Aperiodic, transient |
| Power | \(E = \infty\) | \(0 < P < \infty\) | Periodic, stationary |
| Neither | \(E = \infty\) | \(P = \infty\) | e.g., \(x(t) = t\) |
Signals can undergo two types of transformations:
Consider: [ y(t) = 2 x(3t - 1) + 1 ] 1. Time compression: ( x(3t) ) compresses the signal. 2. Time shift: ( x(3t - 1) ) shifts it to the right by 1 unit. 3. Amplitude scaling: ( 2 x(3t - 1) ) amplifies the signal. 4. Amplitude shift: ( +1 ) shifts it upward.